A202321 -id:A202321 - cp's OEIS frontend

A202320 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.

1, 8, 4, 1, 4, 0, 5, 6, 6, 0, 4, 3, 6, 9, 6, 0, 6, 3, 7, 8, 4, 6, 6, 0, 4, 6, 5, 8, 0, 1, 2, 4, 8, 6, 1, 0, 6, 0, 5, 0, 3, 7, 1, 3, 1, 4, 3, 7, 7, 6, 3, 9, 6, 6, 9, 5, 6, 4, 8, 5, 0, 0, 8, 9, 5, 4, 8, 1, 8, 4, 0, 8, 1, 2, 1, 8, 3, 1, 7, 0, 0, 0, 5, 1, 0, 3, 4, 5, 6, 7, 1, 6, 9, 1, 3, 0, 4, 0, 1

Offset: 1

Clark Kimberling, Dec 16 2011

For many choices of u and v, there is just one x < 0 and one x > 0 satisfying u*x + v = exp(x). Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... least x.....greatest x
1.... 2.... A202320.... A202321
1.... 3.... A202324.... A202325
2.... 1.... ..(x=0).... A202343
3.... 1.... ..(x=0).... A202344
2.... 2.... A202345.... A202346
1.... e.... A202347.... A104689
e.... 1.... ..(x=0).... A202350
3.... 0.... A202351.... A202352
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z = g(u,v) an implicit surface of f.
For an example related to A202320, take f(x,u,v) = u*x + v - exp(x) and g(u,v) = a nonzero solution x of f(x,u,v) = 0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
The solution for u*x + v = exp(x) is -LambertW(-1/(u*exp(v/u))) - v/u. - Andrea Pinos, Sep 14 2023

			x < 0:  -1.841405660436960637846604658012486...
x > 0:   1.1461932206205825852370610285213682...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Wikipedia, Lambert W function. Applications
Index entries for transcendental numbers.

Cf. A202322, A202348.

(* Program 1:  A202320 and A202321 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.9, -1.8}, WorkingPrecision -> 110]
RealDigits[r]  (* A202320 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A202321 *)
(* Program 2: implicit surface of u*x+v=e^x *)
f[{x_, u_, v_}] := u*x + v - E^x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 2, 4}, {u, 2, 4}];
ListPlot3D[Flatten[t, 1]] (* for A202320 *)
RealDigits[-ProductLog[-1/E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *)

solve(x=-2, -1, x + 2 - exp(x)) \\ Michel Marcus, Dec 30 2018

lambertw(-exp(-2))-2 \\ Charles R Greathouse IV, Feb 03 2025

Equals -LambertW(-exp(-2)) - 2. - Vaclav Kotesovec, Jan 09 2014

Equals 2 - A202348. - Jianing Song, Dec 30 2018

A226572 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 + log(f(k-1)) for k>1.

Original entry on oeis.org

3, 1, 4, 6, 1, 9, 3, 2, 2, 0, 6, 2, 0, 5, 8, 2, 5, 8, 5, 2, 3, 7, 0, 6, 1, 0, 2, 8, 5, 2, 1, 3, 6, 8, 2, 5, 2, 8, 8, 8, 6, 6, 2, 0, 4, 6, 1, 8, 2, 4, 8, 8, 4, 2, 6, 0, 3, 4, 6, 1, 9, 2, 9, 1, 2, 8, 6, 7, 7, 5, 1, 6, 3, 9, 8, 7, 5, 4, 8, 8, 7, 0, 7, 7, 4, 3

Offset: 1

Clark Kimberling, Jun 12 2013

Let h(x) be the greater of the two solutions of s + log(s) = x; then this sequence represents h(2). The function h(x) is plotted by the Mathematica program. [This comment is wrong. A226571 = LambertW(exp(2)) = 1.5571455989976... is the unique root of the equation s + log(s) = 2. Equation s - log(s) = 2 does have two roots, but they are s = -LambertW(-1,-exp(-2)) = 3.14619322062... (this sequence) and s = -LambertW(-exp(-2)) = 0.158594339563... (A202348, not A226571). - Vaclav Kotesovec, Jan 09 2014]

Apart from the first digit the same as A202321. - R. J. Mathar, Jun 15 2013

			2 + log 2 = 2.693147...
2 + log(2 + log 2) = 2.990710...
2 + log(2 + log(2 + log 2)) = 3.095510...
limit(f(n)) = 3.14619322062...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A202321, A202348, A226571, A226573, A226574.

Cf. also A052526.

f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[2, 200]][[1]]  (* A226571 *)
d2 = RealDigits[g[2, 200]][[1]]  (* A226572 *)
s /. NSolve[s - Log[s] == 2, 200]  (* both constants *)
h[x_] := s /. NSolve[s - Log[s] == x]
Plot[h[x], {x, 1, 3}, PlotRange -> {0, 1}] (* bottom branch of h *)
Plot[h[x], {x, 1, 3}, PlotRange -> {1, 5}] (* top branch *)

default(realprecision, 100); solve(x=3, 4, x - log(x) - 2) \\ Jianing Song, Dec 30 2018

Equals -LambertW(-1, -exp(-2)) = A202321 + 2. - Vaclav Kotesovec, Jan 09 2014

Definition revised by N. J. A. Sloane, Dec 09 2017

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A202320 Decimal expansion of x < 0 satisfying x + 2 = e^x, negated.

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A226572 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 + log(f(k-1)) for k>1.

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